Continuum robot control methods and apparatus

ABSTRACT

A continuum robot having at least two independently manipulateable bendable section for advancing the robot through a passage, without contacting fragile elements within the passage, wherein the robot incorporates control algorithms that enable the continuum robot to operate and advance into the passage, as well as the systems and procedures associated with the continuum robot and said functionality.

CROSS REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.16/029,461, filed on Jul. 6, 2018, which claims priority from U.S.Provisional Patent Application No. 62/533,466 filed on Jul. 17, 2017, inthe United States Patent and Trademark Office, the disclosure of whichis incorporated herein in its entirety by reference.

FIELD OF THE DISCLOSURE

The present disclosure relates to a control system of a continuum robot.More particularly, the present disclosure is directed toward methods,systems and apparatus for a continuum robot configured withindependently manipulateable bendable section for advancing the robotthrough a passage, without contacting fragile elements within thepassage.

BACKGROUND OF THE DISCLOSURE

A continuum robot includes a plurality of bending sections having aflexible structure, wherein the shape of the continuum robot iscontrolled by deforming the bending sections. The robot mainly has twoadvantages over a robot including rigid links. The first advantage isthat the continuum robot can move along a curve in a narrow space or inan environment with scattered objects in which the rigid link robot mayget stuck. The second advantage is that it is possible to operate thecontinuum robot without damaging surrounding fragile elements becausethe continuum robot has intrinsic flexibility.

As such, the detection of an external force, which is required for therigid link robot, is unnecessary. Taking advantages of thesecharacteristics, it is expected that the continuum robot could beadvantageously applied in the medical field, such as an endoscope sheathand a catheter, and to a robot for hazardous situations, such as arescue robot. United States Patent Publication No. 2012/279 to BelsonAmir (hereafter “PL₁”) describes a control method for enabling acontinuum robot, which is used as an endoscope, for advancement into aspace.

However, with the method taught in PL₁, every pair of adjacent bendingsections are controlled so that the bending shape of a leading sectionbecomes the bending shape of the following section as an endoscope baseadvances, and thereby the shape is continuously propagated, which maylead to unwanted contact with fragile elements as the endoscope isadvanced.

Furthermore, when the continuum robot is used as an endoscope by settingan image-capturing device at the most distal end of the continuum robot,a motion of temporarily stopping the base and looking around (which isreferred to as a “look-around motion”) is performed. However, thecontrol method of continuously propagating the bending posture of themost distal end to the following bending section by the length of thebending section is applied to the look-around motion, the look-aroundmotion of the most distal end is propagated to the following bendingposture, and the continuum robot becomes more likely to contact elementsin a small and narrow space. Thus, the method has problems in that thecontinuum robot cannot advance into a small and narrow space as frictionincreases due to increase of a normal force with an obstacle and thecontinuum robot becomes more likely to break.

Accordingly, it would be particularly beneficial to devise methods,systems and apparatus which would allow for advancement of a medicaldevice without contacting elements, all the while allowing forlook-around functionality, as well as other medical functions desirablein a medical diagnosis, probing or surgical setting.

SUMMARY

Thus, to address such exemplary needs in the industry, the presentlydisclosed apparatus teaches a robotic apparatus comprising: a continuumrobot including a plurality of bending sections including a distalbending section and a proximal bending section wherein each of thebending sections are bent by at least one wire; a driver that drives thewire; a controller that controls a driving amount of the wire; and abase affixed to the continuum robot and capable of moving the continuumrobot, wherein, when a base moves the continuum robot a displacementvalue, the distal bending section performs a rotational motion, and anangle (ζt) of the rotational motion is 360 degrees or more, and thecontroller controls the proximal bending section so as to follow thedistal bending section while preventing the proximal bending sectionfrom performing a rotational motion of 360 degrees or more, based on abending state of the distal bending section at a time when the distalbending section finishes the rotational motion.

In various embodiments, the robot apparatus controller may calculates anangle (ζt′) that is 0 degrees or more and 360 degrees or less and theangle (ζt′) has a same phase as the angle (ζt) of the rotational motion,and performs bending control of the proximal bending section based onthe calculated angle (ζt′).

In other embodiment, the robotic apparatus controller may performbending control of the proximal bending section based on a valueobtained by calculating an angle (ζt′) obtained by using the followingformulas using the angle (ζt) of the rotational motion of the distalbending section

ζt′=ζt mod 2π(ζt>2π)

ζt′ =ζt mod -2π(ζt<−2π).

In yet additional embodiments, the robotic apparatus further provides,wherein, regarding the proximal bending section, the controllercalculates an angle (ζt″) that is −180 degrees or more and less than 180degrees and that has a same phase as the angle (ζt) of the rotationalmotion, and performs bending control of the proximal bending sectionbased on ζt″.

In additional embodiments, the controller performs bending control ofthe proximal bending section based on a value obtained by calculating anangle ζt″ obtained by using the following formulas using the angle (ζt)of the rotational motion of the distal bending section

ζt′=ζt mod 2π(ζt>2π)

ζt′=ζt mod −2π(ζt<−2π)

ζt″=−π+ζt′ mod π(ζt′>π)

ζt″=π+ζt′ mod −π(ζt′<−π).

In another embodiment of the subject robotic apparatus, the controllercalculates an angle ζt″′ obtained by using the following formulas usingthe angle (ζt) of the rotational motion of the distal bending section

ζt′=ζt mod 2π(ζt>2π)

ζt′=ζt mod −2π(ζt<−2π)

ζt″=−π+ζt′ mod π(ζt′>π)

ζt″=π+ζt′ mod −π(ζt′<−π)

ζt′″=ζt″−π(π/2<ζt″<π)

ζt′″=ζt″+π(−π<ζt″<−π/2),

calculates an angle θ′ obtained by using the following formula using abending angle θ of the distal bending section, θ′=−θ, and bends theproximal bending section to be in a state in which the proximal bendingsection is bent at the bending angle 0′ and rotated by the angle ζt′″ ofthe rotational motion.

In yet another embodiment, the robotic apparatus discloses wherein,regarding the proximal bending section, when the distal bending sectionperforms a rotational motion while the predetermined base displacementchanges by a predetermined value and the angle (ζt) of the rotationalmotion is 360 degrees or more, regarding the proximal bending section,an angle (ζt″) that is −180 degrees or more and less than 180 degreesand that has a same phase as the angle of the rotational motion iscalculated, and bending control of the proximal bending section isperformed based on ζt″.

In various embodiments, the robotic apparatus provided: wherein,regarding the proximal bending section, the controller performs bendingcontrol based on bending control of the distal bending section during aperiod in which the base displacement changes by a predetermined value(Δz′).

In other embodiments, the robotic apparatus provided: wherein, when thebase has the displacement value, the controller determines whether ornot the distal bending section performs a rotational motion whose angle(ζt) is 360 degrees or more.

In yet other embodiments, the robotic apparatus teaches, wherein thedistal bending section includes two independent bending sections.

Another embodiment of the subject robot includes further comprising amedial bending section wherein the medial bending section is bent by atleast one wire.

The subject disclosure also teaches a continuum robot including aplurality of bending sections including a distal bending section and aproximal bending section wherein each of the bending sections are drivenby at least one wire; driving means that drives the wire; control meansthat controls a wire driving amount from a bending angle and arotational angle of the continuum robot; and base control means that iscapable of mounting the continuum robot and moving the continuum robot,wherein the control means includes a reference table of the bendingangle θf and the rotational angle ζf at the base displacement, and tablerewriting means that rewrites the reference table in accordance with abending angle ζt and a rotational angle ζt of a most distal bendingsection, and wherein the table rewriting means calculates a follower ζf′as a command value of the rotational angle ζf of a bending sectionexcluding the most distal bending section

ζf′=ζt mod 2π(ζt>2π)

ζf′=ζt mod −2π(ζt>2π).

In other embodiments, the continuum robot control means provides:wherein the table rewriting means calculates a follower ζf″ as thecommand value of the rotational angle ζf of the bending sectionexcluding the most distal bending section

ζf′=−π+ζt′ mod π(ζt′>π)

ζf′=π+ζt′ mod −π(ζt′>−π).

In yet further embodiments, the subject robot provides: wherein thetable rewriting means calculates a follower Cr as the command value ofthe rotational angle ζf of the bending section excluding the most distalbending section

ζf″′=ζt″−π(π/2<ζt″<π)

ζf″′=ζt″+π(−π<ζt″<−π/2).

These and other objects, features, and advantages of the presentdisclosure will become apparent upon reading the following detaileddescription of exemplary embodiments of the present disclosure, whentaken in conjunction with the appended drawings, and provided paragraphs

BRIEF DESCRIPTION OF THE DRAWINGS

Further objects, features and advantages of the present invention willbecome apparent from the following detailed description when taken inconjunction with the accompanying figures showing illustrativeembodiments of the present invention.

FIG. 1 illustrates a kinematic model of the subject continuum robot,according to one or more embodiment of the subject apparatus, method orsystem.

FIG. 2 provides a detailed illustration of the subject continuum robot,according to one or more embodiment of the subject apparatus, method orsystem.

FIG. 3 is a top perspective view of the subject continuum robot,according to one or more embodiment of the subject apparatus, method orsystem.

FIG. 4 illustrates an exemplary procedure of a leader following controlutilized in the subject continuum robot, according to one or moreembodiment of the subject apparatus, method or system.

FIGS. 5(a) and 5(b) provide control systems for the subject continuumrobot, according to one or more embodiment of the subject apparatus,method or system.

FIG. 6 illustrates the control system for the subject continuum robot,according to one or more embodiment of the subject apparatus, method orsystem.

FIG. 7 is a block diagram illustrating the control system of the subjectcontinuum robot, according to one or more embodiment of the subjectapparatus, method or system.

FIGS. 8(a) through 8(d) illustrate a simulation for controlling thesubject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

FIGS. 9(a) through 9(d) provide a simulation for controlling the subjectcontinuum robot, according to one or more embodiment of the subjectapparatus, method or system.

FIG. 10 illustrates a control system for controlling the subjectcontinuum robot, according to one or more embodiment of the subjectapparatus, method or system.

FIGS. 11(a) through 11(d) illustrate a simulation for controlling thesubject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

FIGS. 12(a) through 12(d) illustrate a simulation for controlling thesubject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

FIGS. 13(a) and 13(b) illustrate control systems for controlling thesubject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

FIGS. 14(a) through 14(d) illustrate a simulation result for controllingthe subject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

FIGS. 15(a) through 15(d) illustrate a simulation result for controllingthe subject continuum robot, according to one or more embodiment of thesubject apparatus, method or system.

Throughout the Figures, the same reference numerals and characters,unless otherwise stated, are used to denote like features, elements,components or portions of the illustrated embodiments. In addition,reference numeral(s) including by the designation “′” (e.g. 12′ or 24′)signify secondary elements and/or references of the same nature and/orkind. Moreover, while the subject disclosure will now be described indetail with reference to the Figures, it is done so in connection withthe illustrative embodiments. It is intended that changes andmodifications can be made to the described embodiments without departingfrom the true scope and spirit of the subject disclosure as defined bythe appended paragraphs.

DETAILED DESCRIPTION OF THE DISCLOSURE

In the subject disclosure, Applicant will first detail the mechanism ofa continuum robot, followed by the control algorithms that enable thecontinuum robot to operate and advance into a path, as well as thesystems and procedures associated with the continuum robot and saidfunctionality.

First Embodiment

FIGS. 5(a) and 5(b) are graphs representing a table of a bending anglecommand with respect to a base displacement for the robot. When anoperator issues a bending command angle aB and a rotation command angleab at a base displacement a, the bending angle command for a followermay be automatically generated so that the bending angle becomes cD andthe rotational angle becomes cd at a base displacement c. This will bereferred to as a first leader following control method. Moreover, if therotational angle ab exceeds 2πrad, 2nπrad is subtracted from therotational angle command as illustrated in FIG. 6. In the presentembodiment, this will be referred to as a second leader followingcontrol method.

FIG. 7 shows a block diagram of the subject continuum robot, accordingto one or more embodiment of the subject apparatus, method or system.Here, P denotes a control target, FTL denotes a first leader followingcontrol algorithm, θ_(tref) and ζ_(tref) denote a bending angle commandvalue and a rotational angle command value for the most distal end,θ_(fref) and ζ_(fref) denote a bending angle command vector and arotational angle command vector, Z_(b) denotes a base displacementcommand value, and the block f denotes a second leader following controlalgorithm. The block K represents kinematic calculation and calculates awire driving amount from the bending and rotational angle commandvalues. Details and control results obtained by simulations will bedescribed below.

FIG. 1 illustrates a continuum robot 100 that is capable of a pluralityof bends, with FIG. 2 providing an enlarged view of a first bendingsection 102 at the proximal end 104 of the robot 100.

In the continuum robot 100, wires 111, 112 and 113 are connected toconnection portions 121, 122 and 123, respectively, found on an end disc160 found at the distal end 106 of each bending section 102, Wherein theposture of the bending section 102 is controlled by pushing and pullingthe wires 111 to 113 by using actuators 130 to 132 disposed in a robotbase 140.

Moreover, the robot base 140 of the continuum robot 100 is disposed on abase stage (not shown) and can be moved by the base stage in thelongitudinal direction. Thus, it is possible to advance and retard therobot 100 into a target structure by advancing and retarding the basestage.

A controller (not shown) indicates a driving amount to the base stageand the actuators 130 to 132. Throughout this disclosure, the controllermay also be described or eluded to as a control system. The controllermay include dedicated hardware including a field-programmable gate array(“FPGA”) and the like; or may be a computer including a storage unit, awork memory, and a central processing unit (“CPU”). In the case wherethe controller is a computer, the storage unit may store a softwareprogram corresponding to an algorithm of the control system (describedbelow) and the central processing unit expands the program in the workmemory, executes the program line by line, and thereby the computerfunctions as the controller. In either case, the controller iscommunicably connected to the base stage and the actuators 130 to 132,and the controller send signals representing the driving amount andconfiguration to these control targets.

The continuum robot 100 includes wire guides 161 to 164 situatedthroughout each bending section 102, for guiding the wires 111, 112 and113, and for providing structural integrity to the bending section 102.The wire guides 161 to 164 each contain a wire through 150-153 for eachwire 111-113. For ease of illustration, FIG. 2 only depicts the wirethrough 150-153 for a single wire 111. Alternatively, a method ofdiscretely arranging the plurality of wire guides, a continuum robot 100having a bellows-like shape or a mesh-like shape may be utilized,wherein the wire guides 161-164 are fixed to their respective wires111-113.

With respect to FIGS. 1 and 2, the definitions of symbols are asfollows: l_(d)=the length of the central axis a bending section;θ_(n)=the bending angle of the distal end; ζ_(n)=the rotational angle ofthe distal end; ρ_(n)=the radius of curvature of a bending section.

In various embodiments, the wires 111-113 may be referred to as wires a,b, and c, counterclockwise in the in the xy plane; and the drivingdisplacements of the wires for driving the n-th bending section aredenoted by l_(pna), l_(pnb), and l_(pnc). As illustrated in FIG. 3, thewires a-c are disposed at the vertices of an equilateral triangle whoseside has a length r_(s). The phase angle θ_(n) is an angle thatdetermines the wire arrangement for driving the n-th bending section. Inthe present embodiment, ζ₁=0.

Based on the following assumptions, the kinematics of the continuumrobot 100 may be derived: 1. In each bending section, the wires deformwith a constant curvature; 2. Twisting deformation of the wires is notconsidered; 3. The wires do not deform in the longitudinal direction; 4.Friction between the wire guides and the wires is not considered.

With these assumptions in mind, we define the following symbols. First,the relationships between the driving displacements l_(p1a,) 1_(p1b),and l_(p1c) of the wires a, b, and c and the bending angle θ₁ and therotational angle ζ₁ of the first bending section are expressed asfollows.

$\begin{matrix}{{l_{p\; 1a} = {\frac{r_{s}}{\sqrt{3}}\cos\;\zeta_{1}\theta_{1}}}{l_{p\; 1b} = {\frac{r_{s}}{\sqrt{3}}{\cos\left( {\frac{\pi}{6} + \zeta_{1}} \right)}\theta_{1}}}{l_{p\; 1c} = {\frac{r_{s}}{\sqrt{3}}{\cos\left( {\frac{\pi}{6} - \zeta_{1}} \right)}\theta_{1}}}} & (1)\end{matrix}$

Next, for the continuum robot 100 including the plurality of bendingsections 102, the relationships between the driving displacementsl_(pna), l_(pnb), and l_(pnc) of the wires a, b, and c and the bendingangle θ_(n) and the rotational angle ζ_(n) of the distal end areobtained. The phase angle of the wires for driving the n-th bendingsection is expressed as follows, where e denotes the number of bendingsections.

$\begin{matrix}{\xi_{n} = {\frac{120}{e}n}} & (2)\end{matrix}$

Thus, the driving displacements l_(pna), l_(pnb), and l_(pnc) of thewires of the n-th bending section are expressed as follows.

$\begin{matrix}{{l_{p\;{na}} = {\frac{r_{s}}{\sqrt{3}}\cos\;\left( {\zeta_{n} - \xi_{n}} \right)\theta_{n}}}{l_{p\;{nb}} = {\frac{r_{s}}{\sqrt{3}}{\cos\left( {\frac{\pi}{6} + \zeta_{n} - \xi_{n}} \right)}\theta_{n}}}{l_{p\;{nc}} = {\frac{r_{s}}{\sqrt{3}}{\cos\left( {\frac{\pi}{6} - \zeta_{n} + \xi_{n}} \right)}\theta_{n}}}} & (3)\end{matrix}$

The subject disclosure now provides details regarding a leader followingcontrol system. As illustrated in FIG. 4, leader following control is amethod of performing control so that a following bending section passesthrough the same path as a path through which a bending section at themost distal end passes. Thus, the continuum robot 100 can advancethrough a space, while retaining the original bend initiated to avoidcontact, as well as initiating new bends to avoid up and comingobstacles. In the leader following control, the path need not bedetermined beforehand. It is sufficient that the bending angle of themost distal end is continuously propagated to the following bendingsection by a bending section length. With this method, an operator canperform the leader following control in real time by only givingcommands with respect to the bending angle of the most distal end of therobot 100 while advancement of the base is promoted by using a joystickor the like.

In further detailing FIG. 4, the horizontal line represents the passageof time as the robot 100 is advanced into the desired space, with thearrows signifying a progression of time. As the base 140 is advanced,the first bending section 102 takes the shape of the first desired bend(dotted line) 190. Further progression of the base 140 depicts thesecond bending section 104 taking the shape of the first desired bend190, while the first bending section 102 takes the shape of the seconddesired bend 192. Additional bending sections and desired bends arecontemplated and further claimed herein.

a) First Leader Following Control Method

FIGS. 5(a) and 5(b) are graphs in which the horizontal axes representthe base displacement z_(b) and the vertical axes respectively representthe bending angle θ and the rotational angle ζ. The thinner broken linerepresents a bending command given by an operator to the distal bendingsection, and the thicker broken line represents a bending command to thefollowing bending section (follower). When an operator issues the distalbending command angle aB and the rotation command angle ab at the basedisplacement a, the bending angle for the follower may be automaticallygenerated so that the bending angle and the rotational anglerespectively become cD and cd at the base displacement c. Here, the basedisplacement c is determined so that the distance ac becomes the bendingsection length l_(d). Then, the bending angle command of the follower isstored in the storage unit of the control processing device and read outin accordance with the base displacement. When the number of bendingsections is two or more, it is possible to obtain bending angle commandvalues for all bending sections by substituting the follower section inthe above description with the distal end and by continuously performingthe processing.

However, with this command value, when the base displacement is a or c,the bending and the rotational angles of the follower do not change, andthe bending and rotational angle commands rise at the base displacementc, and therefore the continuum robot shows an abrupt behavior.Therefore, in the present disclosure, the bending angle command of thefollower is interpolated so as to connect the point a and the point Dand the rotational angle command is interpolated so as to connect thepoint a and the point d. The solid line in FIG. 5 represents theinterpolated bending angle command for the follower. In the presentdisclosure, the angle command generation algorithm described in thissection is referred to as a first leader follower control method.

b) Second Leader Following Control Method

When the first leader follower control method described in the previoussection is applied to a command value in which the rotational anglecommand ab exceeds 2πrad, the follower performs a rotational motion ofone or more rotations around the z axis as the base advances. It isconsidered that a rotation operation command given by an operator toperform a rotational motion of one or more rotations around the z axisat the base displacement a is, for example, a motion of looking aroundat the position by using an image-capturing device disposed at thedistal end of the robot. When the base advances after the look-aroundmotion, the follower need not perform this motion. This is because it isdifficult to dispose an image-capturing device or the like in thefollower section, because the follower section is continuous with theleading section. Moreover, if this motion is performed in a path in anarrow and small space surrounded by obstacles, the motion range of thecontinuum robot becomes larger and contact with a surrounding elementsis more likely to occur. Contact with surrounding elements is even morelikely to occur when the length of the follower is large relative to thelength of the leader.

Therefore, in the present embodiment, a second leader following controlalgorithm illustrated in FIG. 6 is created. As in FIG. 5(b), the solidline, the thick broken line, and the broken line respectively representa bending angle command after interpolation, a bending angle commandbefore interpolation, and a command by an operator. In the graphprovided in FIG. 6, n denotes a natural number. The second leaderfollower control algorithm calculates a rotational angle command cd′ asa rotation command for the second follower by subtracting a rotationalmotion of n rotations from the rotation command for the leader; and theninterpolates a rotational angle command so as to connect the point a andthe point d′ in the same way as in the first follower control method. Inthis case, the rotational angle command cd′ for the follower can becalculated from the following formulas.

ζ_(cd′)=ζ_(cd) mod2π(ζ_(cd)>2π)

ζ_(cd′)=ζ_(cd) mod−2π(ζ_(cd)<−2π)  (4)

Here, ζ_(cd) and ζ_(cd′) respectively denote the rotational anglecommands cd and cd′. The symbol “mod” represents modular arithmetic, andthe sign of the solution is the same as the divisor. Thus, it ispossible to prevent a rotational motion of one or more rotations of thedistal end of the robot around the z axis from propagating to thefollower.

FIG. 7 illustrates a block diagram illustrating the control system 170of the subject continuum robot 100, according to one or more embodimentof the subject apparatus, method or system. The control system 100comprises a control target 172, a first leader following controlalgorithm 174, wherein θ_(tref) and ζ_(tref) denote a bending anglecommand value and a rotational angle command value of the distal end,and θ_(fref) and ζ_(fref) represent a bending angle command vector and arotational angle command vector. In addition, Z_(b) denotes a basedisplacement command value, with the control system 170 furtherutilizing the modular arithmetic 178 shown in Formula (4). The kinematiccalculation 178, described in the first chapter, calculates a wiredriving amount from the bending and rotational angle command values.

Simulation

All simulations are performed by using one or more embodiment of theleader follower control system described in the previously. Thesimulation is performed on a continuum robot that includes two bendingsections each having a bending section length of 0.01 m.

FIGS. 8(a) to 8(d) are simulations by stick diagrams illustration,wherein a stepwise manner, the postures when control using the secondleader following control algorithm is performed until the base advancesto 0.01 m. The solid line represents the shape of the robot, the blackdot represents the distal end of each bending section, and the thinsolid line represents the locus of the distal end of each bendingsection. The initial posture shown in FIG. 8(a) is expressed as follows.

${z_{b} = 0},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{15}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}0 & {\frac{810}{180}\pi}\end{bmatrix}}$

In the initial posture, the second bending section has already performeda look-around motion of two rotations around the z axis. By performingcontrol using the second leader follower control algorithm, the bendingangle θ₁ of the final posture is equal to the bending angle θ₂ of theinitial posture, and the rotational angle θ₁ of the final posture is anangle calculated from the bending angle ζ₂ of the initial posture byperforming modular arithmetic as follows.

${z_{b} = 0.01},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{45}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{90}{180}\pi} & {\frac{810}{180}\pi}\end{bmatrix}}$

It can be seen that, due to the modular arithmetic, the first bendingsection is not affected by the look-around motion of the second bendingsection and the leader follower control is performed smoothly. Forcomparison, FIG. 9(a) to 9(d) illustrate responses due to the firstleader follower control algorithm. The look-around motion of the secondbending section is directly propagated to the motion of the firstbending section, and therefore the first bending section performs arotational motion of two rotations as the base advances. Thus, it can beseen that not only the first bending section but also the second bendingsection rotate by a large amount as the base advances, and contact withexternal environment is likely to occur in a narrow and small space.

Second Embodiment

In the first Embodiment, although a rotational motion or one turn ormore is subtracted from the most distal end, the following sectionperforms a rotational motion in the same direction as the distal end.However, it is possible to reduce the rotation amount by rotating thefollowing section in a direction opposite to the direction of theleading section. Therefore, in the present embodiment, a third leaderfollower control algorithm, as illustrated in FIG. 10, will bedescribed.

As in the first embodiment, the solid line, the thick broken line, andthe broken line respectively represent a bending angle command afterinterpolation, a bending angle command before interpolation, and acommand by an operator. The third leader follower control algorithmperforms the second leader follower control algorithm; and thencalculates a rotational angle command cd″ as a rotation command for thefollower by adding or subtracting 2π to or from the rotation command forthe leader. Then, in the same way as in the second leader followercontrol method, the algorithm interpolates the rotational angle commandso as to connect the point a and the point d″. In this case, therotational angle command cd″ for the follower can be calculated from thefollowing formulas.

ζ_(cd″)=−π+ζ_(cd′) modπ (ζ_(cd′>π))

ζ_(cd″)=π+ζ_(cd′) mod . . . π (ζ_(cd′)< . . . π)  (5).

Thus, when compared with the second leader follower control algorithm,it is possible to further subtract a rotation amount of a half rotation.

Simulation is performed by using the third leader follower controlsystem. FIGS. 11(a) to 11(d) are stick diagrams illustrating, in astepwise manner, the postures when control using the third leaderfollower control algorithm is performed and the base advances by 0.01 m.The solid line represents the shape of the robot, the black dotrepresents the distal end of each bending section, and the thin solidline represents the locus of the distal end of each bending section. Theinitial posture shown in FIG. 11(a) is expressed as follows.

${z_{b} = 0},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{15}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}0 & {\frac{330}{180}\pi}\end{bmatrix}}$

By performing control using the third leader follower control algorithm,the bending angle θ₁ of the final posture is equal to the bending angleθ₂ of the initial posture, and the rotational angle ζ₁ of the finalposture is an angle calculated from the bending angle ζ₂ of the initialposture by performing modular arithmetic as follows.

${z_{b} = 0.01},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{45}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{- 30}{180}\pi} & {\frac{330}{180}\pi}\end{bmatrix}}$

It can be seen that, due to the modular arithmetic, the leader followercontrol is performed smoothly. For comparison, FIGS. 12(a) to 12(d)illustrate responses due to the second leader follower controlalgorithm. In the motion of the first bending section, the first bendingsection rotates in the same direction as the motion of the secondbending section, and therefore the rotation amount due to theadvancement of the base is large. Thus, it can be seen that, not onlythe first bending section but also the second bending section rotate bya large amount as the base advances, and contact with externalenvironment is likely to occur in a narrow and small space.

Third Embodiment

In the first and second embodiments, the rotation amount of the followersection is reduced by operating the rotational angle command. However,it is possible to further reduce the rotation amount by operating thebending angle command for the following section. Therefore, in thepresent embodiment, a fourth leader follower control algorithm,illustrated in FIG. 13, will be described.

As in the first embodiment, the solid line, the thick broken line, andthe broken line respectively represent a bending angle command afterinterpolation, a bending angle command before interpolation, and acommand by an operator. The fourth leader follower control algorithmperforms the third leader following control algorithm; and thencalculates a rotational angle command cd′″ as a rotation command for thefollower by adding or subtracting π to or from a rotation command forthe leader. Then, in the same way as in the third leader followercontrol method, the algorithm interpolates the rotational angle commandso as to connect the point a and the point d′″. In this case, therotational angle command cd′″ for the follower can be calculated fromthe following formulas.

$\begin{matrix}{{{{{\zeta_{cd}}^{\prime}}^{\prime}}^{\prime} = {{\zeta_{cd}}^{''} - {\pi\mspace{14mu}\left( {\frac{\pi}{2} < {\zeta_{cd}}^{''} < \pi} \right)}}}{{{{\zeta_{cd}}^{\prime}}^{\prime}}^{\prime} = {{\zeta_{cd}}^{''} + {\pi\mspace{14mu}\left( {{- \pi} < {\zeta_{cd}}^{''} < {- \frac{\pi}{2}}} \right)}}}{{\theta_{cD}}^{\prime} = {- \theta_{cD}}}} & (6)\end{matrix}$

Here, θ_(cD′) and −θ_(cD) respectively denote the bending angle commandscD and cD′. Thus, when compared with the third leader follower controlalgorithm, it is possible to further subtract a rotation amount of aquarter rotation.

Simulation is performed by using the fourth leader follower controlsystem. FIGS. 14(a) to 14(d) are stick diagrams illustrating, in astepwise manner, the postures when control using the third leaderfollower control algorithm is performed and the base advances by 0.01 m.The solid line represents the shape of the robot, the black dotrepresents the end of each bending section, and the thin solid linerepresents the locus of the end of each bending section. The initialposture shown in FIG. 14(a) is expressed as follows.

${z_{b} = 0},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{15}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}0 & {\frac{160}{180}\pi}\end{bmatrix}}$

By performing control using the fourth leader follower controlalgorithm, the bending angle θ₁ of the final posture has a sign oppositeto the sign of the bending angle θ₂ of the initial posture, and therotational angle ζ₁ of the final posture is an angle calculated from thebending angle ζ₂ of the initial posture by performing addition andsubtraction as follows.

${z_{b} = 0.01},{\begin{bmatrix}\theta_{1} & \theta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{- 45}{180}\pi} & {\frac{45}{180}\pi}\end{bmatrix}},{\begin{bmatrix}\zeta_{1} & \zeta_{2}\end{bmatrix} = \begin{bmatrix}{\frac{- 20}{180}\pi} & {\frac{160}{180}\pi}\end{bmatrix}}$

Thus, it can be seen that the leader follower control is performedsmoothly. For comparison, FIGS. 12(a) to 12(d) illustrate responses dueto the third leader follower control algorithm. In the motion of thefirst bending section, the rotation amount due to the advancement of thebase is large. Thus, it can be seen that the distal end of the secondbending section shows a locus on an arc, and contact with externalenvironment is likely to occur in a narrow and small space.

As described above, in the embodiments according to the presentdisclosure, when a distal bending section performs a rotational motionwherein the angle (ζcd or ζt) of the rotational motion is 360 degrees ormore, a controller performs bending control so that a proximal bendingsection follows the distal bending section while preventing the proximalbending section from performing a rotational motion of 360 degrees ormore. That is, based on the direction of the distal bending section atthe start and at the end of the rotational motion, the effect ofunnecessary rotation is avoided, and therefore it is possible to reduceor eliminate issues, such as contact with external environment.

Thus, the following advantages are obtained. For example, consider acase wherein advancement of the continuum robot is stopped, the distalportion is rotated, image capturing is performed while looking around byusing an objective lens disposed at the distal end of the continuumrobot, and subsequently the robot is further advanced. In this case, itis considered that the look-around motion does not affect theadvancement of the robot. By performing control according to theembodiments, it is possible to advance the proximal bending section soas to follow the distal bending section without being affected by thelook-around motion.

In the first embodiment, the controller calculates an angle (ζcd′ orζt′) that is 0 degrees or more and less than 360 degrees and that hasthe same phase as the angle of the rotational motion, and performsbending control of the proximal bending section based on the calculatedangle. The aforementioned (4) may be used as the calculation formulas,or other theoretical formulas that provide the same result may be used.When realizing these formulas by using a computer, necessaryapproximation may be used.

In the second embodiment, regarding the proximal bending section, anangle (ζcd″ or ζt″) that is −180 degrees or more and less than 180degrees and that has the same phase as the angle of the rotationalmotion is calculated, and bending control of the proximal bendingsection based on ζcd″ is performed. Thus, the rotational motion of theproximal bending section is limited to 180 degrees or less, andtherefore unnecessary motion of the proximal portion is furthersuppressed. As in the first embodiment, the aforementioned (4) and (5)may be used as the calculation formulas, or other theoretical formulasthat provide the same result may be used. When realizing these formulasby using a computer, necessary approximation may be used.

In the third embodiment, unnecessary rotational motion is suppressed bytaking not only the rotational angle but also the bending direction intoconsideration. That is, the controller calculates an angle ζcd″′obtained from the following formulas obtained by using theaforementioned (4), (5) and (6) using the angle ζcd (or ζt) of therotational motion of the distal bending section. The controllercalculates an angle θcD′ obtained by using the aforementioned formula(6) using the bending angle θcD of the distal bending section. Based onthe values of ζcd″′ and θcD′, the proximal bending section is bent to bein a state in which the proximal bending section is bent by the bendingangle OcD′ and rotated by the angle ζcd″′ of the rotational motion. Asin the third embodiment, the aforementioned (4), (5) and (6) may be usedas the calculation formulas, or other theoretical formulas that give thesame result may be used. When realizing these formulas by using acomputer, necessary approximation may be used.

In the embodiments described above, a case where a rotational motion of360 degrees or more is performed at a predetermined base displacement zbis described. In another embodiment, processing intended as describedabove is applied also to a case where a rotational motion is performedfor a slight change Δz≥0 in the base displacement. That is, whenperforming a rotational motion of an angle ζcd (or ζt) of 360 degrees ormore for Δz, an angle (ζcd″ or ζt″) that is 180 degrees or more and lessthan 180 degrees and that has the same phase as the angle as ζcd (or ζt)is calculated, and the proximal bending section is controlled by usingthe angle ζt″ as a rotational angle command value. Thus, it is possibleto reduce the proximal portion from performing an unnecessary followingmotion and to reduce or eliminate issues such as contact with externalenvironment.

In the embodiments described above, when the distal bending sectionperforms a general bending motion including a rotational motion at acertain base displacement zb, as illustrated in FIGS. 5(a) and 6, theproximal bending section is controlled so that the proximal bendingsection gradually bends while the proximal bending section becomesdisplaced by the length ac. In the other embodiment, when the distalbending section performs a bending motion by a predetermined angle whilethe distal bending section becomes displaced by a very short Δz(≥0), theproximal bending section gradually bends by an angle corresponding tothe predetermined angle while the proximal bending section becomesdisplaced for a distance larger than Δz.

The controller determines whether or not the distal bending sectionperforms a rotational motion of an angle (ζt) that is 360 degrees ormore for a predetermined change Δz (≥0) of the base displacement zb.Such control may be determined based on the fact that the distal bendingsection has actually performed a rotational motion or may be determinedbased on the fact that a driving amount or a control value forperforming the rotational motion has been input to the actuators 130 to132. Alternatively, it may be determined when the driving amount or thecontrol value is calculated by the controller.

It is possible to apply control according to the present disclosure notonly in cases where the control described above is performed, but alsoin situations where the bending motion or the rotational motion of theproximal portion is performed, for example, depending on the history orthe locus of the bending motion or the rotational motion of the distalportion.

In the embodiments described above, bending control of the distal end ofthe robot is determined based on the bending motion or the rotationalmotion of the distal portion. However, embodiments of the presentdisclosure are not limited to this embodiment. For example, in a casewhere a continuum robot includes three or more bending sections, twobending sections in the distal portion may perform a rotational motionin synchronism or the second most distal bending section may perform arotational motion. The control algorithms described above may be appliedto bending control of a bending section on the proximal side of the twobending sections. Such an embodiment is also included in the embodimentsof the present disclosure. The point is that, when there are two bendingsections in the distal portion, based on the rotational motion ofbending sections in the distal portion, the aforementioned controlalgorithm may be applied to a bending section on the proximal side.

The controller may determine that the control algorithm according to anembodiment of the present disclosure is applied based on various set(s)of information. In the case where the control algorithm is not applied,for example, control shown in the graph illustrated in FIG. 5(b) isperformed.

1. A robotic apparatus comprising: a continuum robot including aplurality of bending sections including a distal bending section and aproximal bending section wherein each of the bending sections are bentby at least one wire; a driver that drives the wire; a controller thatcontrols a driving amount of the wire; and a base affixed to thecontinuum robot and capable of moving the continuum robot, wherein, whenthe base moves the continuum robot a displacement value, the distalbending section performs a rotational motion, and an angle (ζt) of therotational motion is greater than 0 degrees, and the controller controlsthe proximal bending section so as to follow the distal bending sectionwhile preventing the proximal bending section from performing arotational motion of 360 degrees or more, based on a bending state ofthe distal bending section at a time when the distal bending sectionfinishes the rotational motion.
 2. The robotic apparatus according toclaim 1, wherein the controller calculates an angle (ζt′) that is 0degrees or more and 360 degrees or less and the angle (ζt′) has a samephase as the angle (ζt) of the rotational motion, and performs bendingcontrol of the proximal bending section based on the calculated angle(ζt′).
 3. The robotic apparatus according to claim 1, wherein thecontroller performs bending control of the proximal bending sectionbased on a value obtained by calculating an angle (ζt′) obtained byusing the following formulas using the angle (ζt) of the rotationalmotion of the distal bending sectionζt′=ζt mod 2π(ζt>2π)ζt′=ζt mod −2π(ζt<−2π).
 4. The robotic apparatus according to claim 1,wherein, regarding the proximal bending section, the controllercalculates an angle (ζt″) that is −180 degrees or more and less than 180degrees and that has a same phase as the angle (ζt) of the rotationalmotion, and performs bending control of the proximal bending sectionbased on ζt″.
 5. The robotic apparatus according to claim 1, wherein thecontroller performs bending control of the proximal bending sectionbased on a value obtained by calculating an angle ζt″ obtained by usingthe following formulas using the angle (ζt) of the rotational motion ofthe distal bending sectionζt=ζt mod 2π(ζt>2π)ζt′=ζt mod −2π(ζt<−2π)ζt″=−π+ζt′ mod π(ζt′>π)ζt″=π+ζt′ mod −π(ζt′<−π).
 6. The robotic apparatus according to claim 1,wherein the controller calculates an angle ζt″′ obtained by using thefollowing formulas using the angle (ζt) of the rotational motion of thedistal bending sectionζ=ζt mod 2π(ζt>2π)ζt′=ζt mod −2π(ζt<−2π)ζt″=−π+ζt′ mod π(ζt′>π)ζt″=π+ζt′ mod −π(ζt′<−π)ζt″′=ζt″−π(π/2<ζt″<π)ζt″′=ζt″+π(−π<ζt″<−π/2), calculates an angle θ′ obtained by using thefollowing formula using a bending angle θ of the distal bending section,θ′=−θ, and bends the proximal bending section to be in a state in whichthe proximal bending section is bent at the bending angle θ′ and rotatedby the angle ζt″′ of the rotational motion.
 7. The robotic apparatusaccording to claim 1, wherein, regarding the proximal bending section,when the distal bending section performs a rotational motion while thepredetermined base displacement changes by a predetermined value and theangle (ζt) of the rotational motion is 360 degrees or more, regardingthe proximal bending section, an angle (ζt″) that is −180 degrees ormore and less than 180 degrees and that has a same phase as the angle ofthe rotational motion is calculated, and bending control of the proximalbending section is performed based on ζt″.
 8. The robotic apparatusaccording to claim 1, wherein, regarding the proximal bending section,the controller performs bending control based on bending control of thedistal bending section during a period in which the base displacementchanges by a predetermined value (Δz′).
 9. The robotic apparatusaccording to claim 1, wherein, when the base has the displacement value,the controller determines whether or not the distal bending sectionperforms a rotational motion whose angle (ζt) is 360 degrees or more.10. The robotic apparatus according to claim 1, wherein the distalbending section includes two independent bending sections.
 11. Therobotic apparatus according to claim 1, further comprising a medialbending section wherein the medial bending section is bent by at leastone wire.